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5. Simplify Algebraic Expressions
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Chapter 4
5. 

Simplify Algebraic Expressions

This lesson focuses on simplifying algebraic expressions, a fundamental skill in algebra. It teaches how to identify like terms in an expression and combine them to make the expression simpler. This skill is particularly useful in various real-world scenarios, such as calculating total costs or distances. For example, if someone is trying to figure out the total distance for a multi-leg journey, they can use these techniques to combine the distances into a single, simpler expression. Understanding how to simplify algebraic expressions can make problem-solving quicker and more efficient, whether you're a student or a professional dealing with data analysis.

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Student Learning Objectives:
  • Use like terms to simplify algebraic expressions
  • Use the Distributive Property to simplify algebraic expressions
9 Theory slides
10 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Simplify Algebraic Expressions
Slide of 9
There are many ways to write the same mathematical idea. Some forms are more complex than others, but in general, simpler expressions express ideas more efficiently and are easier to evaluate. This lesson explains how to write an expression as simply as possible.

Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.

Explore

How Can These Terms Be Grouped?

Group the terms in the table correctly.

Grouping Like Terms

What do these terms have in common? Can we combine terms in the same column? Across columns?
Discussion

Like Terms

Two or more terms in an algebraic expression are like terms if they have the same variable(s) with the same exponent(s). 7xy+2x-3xy-2x^2+x+5x^2 In this expression, there are three sets of like terms — xy-terms, x-terms, and x^2-terms.

All constants are like terms. Like terms in an expression can be combined using addition or subtraction to rewrite the expression in simplest form.
Pop Quiz

Identifying Like Terms

Are the terms are like terms?

Are the terms like terms
Discussion

Reducing the Number of Terms

Once we identify like terms in an expression, we can combine them using addition or subtraction.

Method

Combining Like Terms

In an expression, groups of like terms can be combined to write an equivalent expression with the least possible number of terms. Consider the following expression. 7xy+2x-3xy-2x^2+x+5x^2 There are three steps to follow to combine like terms.

1
Identify Like Terms
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Start by identifying the like terms in the expression. These are terms with the same variables raised to the same power. All constants are like terms.

Identifying Like Terms

This expression has two x-terms, two xy-terms, and two x^2-terms.

2
Grouping Like Terms
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Now that the like terms are identified, use the Commutative Property of Addition to rearrange the expression so that the like terms are grouped together.

Grouping Like Terms

3
Add and Subtract the Coefficients
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Finally, combine the like terms by adding or subtracting the coefficients of the variables as well as adding or subtracting the constants.

Combining Like Terms

Now that all like terms have been combined, the initial expression has been reduced to its simplest form. 7xy+2x-3xy-2x^2+x+5x^2 ⇕ 4xy+3x+3x^2

\\\\|LHS| = |RHS| arg(LHS) = arg(RHS)
Example

The Path to the Museum

The map shows a path from Heichi's house to the museum. The distances are given as algebraic expressions.

Little map

Write an expression in simplest form for the total distance from Heichi's house to the museum.

Hint

Identify the like terms and then combine them.

Solution
To find the total distance from Heichi's house to the museum, we can add the expressions for each part of the path he takes. x+ 3x+7+ 4x-6 To simplify this expression, we can use the Commutative Property of Addition to group the like terms together. In the expression for the total distance, we have constant terms and x-terms, which we can then add or subtract. x+ 3x+ 7+ 4x - 6 Let's simplify!

x+ 3x+ 7+ 4x - 6
CommutativePropAdd

Commutative Property of Addition

x+ 3x+ 4x+ 7 - 6
AddTerms

Add terms

8x+7-6
SubTerm

Subtract term

8x+1

The simplified expression is 8x+1.

Example

Weekend Visitors to the Museum

On weekends, the museum has special admission prices before noon.

Building

Children pay x dollars and adults pay x^2 dollars before noon. After noon, the prices go up to y and y^2 dollars for children and adults, respectively. Heichi kept track of the number of visitors on both Saturday and Sunday in separate tables. The following table shows the number of visitors on Saturday.

Saturday Visitors Sunday Visitors
Children Adults Children Adults
Before Noon 7 12 8 15
After Noon 12 20 3 4

How much money did the museum take in from admissions on Saturday and Sunday in terms of x and y? Simplify the total amount.

Hint
The tickets have different prices depending on the time of day.

Solution
We can start by finding Saturday's ticket revenue by multiplying the number of people who entered by the amount they had to pay. Since the prices change depending on the hour, we can do this in two steps. Let's organize our work in a table.

Saturday Visitors
Children ($) Adults ($) Total ($)
Before Noon 7x 12x^2 7x+ 12x^2
After Noon 12y 20y^2 12y+ 20y^2

Now, we can write the expression for Saturday's revenue by adding the expressions for each part of the day. Saturday's Revenue ($) 7x + 12x^2 + 12y + 20y^2 Let's follow the same steps for Sunday.

Sunday Visitors
Children ($) Adults ($) Total ($)
Before Noon 8x 15x^2 8x+ 15x^2
After Noon 3y 4y^2 3y+ 4y^2

Next, we can write the expression for Sunday's revenue by adding the expressions for each part of the day. Sunday's Revenue ($) 8x+ 15x^2+ 3y+ 4y^2 Finally, we can find the total revenue for the weekend by adding the revenues for both days. Let's use the Commutative Property of Addition to group the like terms so we can combine them.

7x + 12x^2 + 12y + 20y^2 + 8x + 15x^2 + 3y + 4y^2
CommutativePropAdd

Commutative Property of Addition

7x+ 8x + 12y + 3y + 12x^2 + 15x^2 + 20y^2 + 4y^2
AddTerms

Add terms

15x+15y + 27x^2 + 24y^2

The total revenue for the weekend is 15x+15y+27x^2+24y^2 dollars.

Example

A Curious Occurrence in the Souvenir Shop

The souvenir shop at a museum sells three large items.

items from the store. Vase has a price a, plus toy a price of b and painting has a price of c

Three groups of people visited the shop on Saturday.

  • The first group bought three vases, four plush toys, and five paintings.
  • The second group bought six plush toys, returned two vases, and bought one painting.
  • The third group returned one painting and bought one vase.

The exact same thing happened on Sunday. Suppose that the items returned are fully refunded. Write and simplify an expression for how much money the shop made from these groups of people over the weekend.

Hint
Start by writing an expression for Saturday. Then, multiply that expression by 2.

Solution
We start by writing an expression for the amount of money the store made on Saturday. To do this, we multiply the number of each item by its price and add the amounts for each group separately. When an item is bought, we use its price as a positive number. When an item is returned, we use a negative number because the store gives the money back to the customer.

Vases ($) Plush Toys ($) Paintings ($) Total ($)
First Group 3a 4b 5c 3a+ 4b+ 5c
Second Group -2a 6b 1c -2a+ 6b+ 1c
Third Group 1a 0 -1c 1a - 1c

The sum of these expressions gives us the money the store made on Saturday. Saturday's Revenue ($) 3a + 4b + 5c+ 6b - 2a + 1c - 1c + 1a The same thing happened on Sunday. To find the money the store made on both days, we multiply the whole expression by 2. Weekend Revenue ($) 2( 3a + 4b + 5c+ 6b - 2a + 1c - 1c + 1a) We can simplify the expression by combining like terms and then distributing the 2. Let's combine the terms first.

2( 3a + 4b + 5c+ 6b - 2a + 1c - 1c + 1a)
CommutativePropAdd

Commutative Property of Addition

2( 3a - 2a + 1a + 4b + 6b + 5c + 1c - 1c)
AddTerms

Add terms

2(4a -2a + 10b + 6c - 1c)
SubTerms

Subtract terms

2(2a+10b+5c)
Distr

Distribute 2

2*2a + 2*10b + 2*5c
Multiply

Multiply

4a+20b+10c

The total revenue for the weekend is 4a+20b+10c dollars.

Closure

Grouping Like Terms

The beginning of this lesson presented a table to group like terms.

Grouping Like Terms
The column headers identify different types of terms. Since all constant terms are like terms, they are grouped in the first column. The terms in the other columns indicate which variables are to be placed there.

Now we can combine the like terms by adding or subtracting them!



Level 1
Level 2
Level 3
Simplify Algebraic Expressions
Exercise 3.1

Consider the following expression. 8y + 4 + 9y +4y +9 - 6 Adding one set of parentheses makes this expression equivalent to the expression 7(3y+5). Rewrite the expression with these parentheses without rearranging the terms.

Consider the given expression. 8y + 4 + 9y +4y +9 - 6 We want to add one set of parentheses to the given expression so that it is equivalent to 7(3y+5). Let's start by simplifying the expression with parentheses to make it easier to compare. We can use the Distributive Property.

Now let's consider the given expression again. We can discard some possible placements of the parentheses without performing any of the operations. We know that we will not need to add the parentheses in a place where the order of operations makes them irrelevant. For example, we do not need to add parentheses to multiply before adding. (8y ) + 4 + (9y ) + (4y ) +9 - 6 We also want to avoid adding parentheses that would result in multiplying two y-terms together. (8y + 4 + 9 )y +4y +9 - 6 If we distributed y in the expression, we would have a y^2-term. Our equivalent expression does not have any y^2-terms, so we know not to place the parentheses here. Consider the following three options.

  1. 8(y + 4) + 9y +4y +9 - 6
  2. 8y + 4 + 9y +4(y +9) - 6
  3. 8y + 4 + 9y +4(y +9 - 6)

Let's simplify the expressions.

Expression Distribute Simplify
8(y + 4) + 9y +4y +9 - 6 8y + 32 + 9y +4y +9 - 6 21y+35
8y + 4 + 9y +4(y +9) - 6 8y + 4 + 9y +4y +36 - 6 21y+34
8y + 4 + 9y +4(y +9 - 6) 8y + 4 + 9y +4y +36 - 24 21y+56

We can see that the first expression is equivalent to 7(3y+5). Good job!

E
Hint & Answer
S
Solution

Find the difference between the perimeters of the rectangle and the pentagon.

A rectangle and a pentagon

We want to find the difference between the perimeters of the given figures.

First, we need to find the perimeter of each figure. Remember that the perimeter measures the distance around the outside of a two-dimensional figure. To determine the perimeter of a figure, add up all of its side lengths. Let's start by adding the given lengths of the rectangle together. a + 2a+3 = 3a+3 In a rectangle, each pair of sides has the same length. Because of this, we can multiply our expression by 2 to find the whole perimeter. Let's use the Distributive Property.

We have an expression for the perimeter of the rectangle. We are making great progress! Now let's add the given side lengths of the pentagon.

Finally, let's calculate the difference between the perimeters.

Since the difference is 0, the perimeters of the rectangle and the pentagon are equal!

E
Hint & Answer
S
Solution

Simplify Algebraic Expressions

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